PSTAT 5A: Continuous Random Variables

Lecture 9

Narjes Mathlouthi

2025-07-23

Welcome to Lecture 9

Continuous Random Variables

From discrete jumps to smooth curves: modeling the continuous world

Today’s Learning Objectives

By the end of this lecture, you will be able to:

  • Distinguish between discrete and continuous random variables (Section 3)
  • Understand probability density functions (PDFs) and their interpretation (Section 4)
  • Work with cumulative distribution functions (CDFs) for continuous variables
  • Calculate probabilities using areas under curves
  • Compute expected values and variances for continuous distributions
  • Work with common continuous distributions (Uniform, Normal, Exponential)
  • Apply the Central Limit Theorem
  • Use python to work with continuous distributions

Review: Discrete vs Continuous

Discrete Random Variables

  • Countable values (can list them)
  • Gaps between possible values
  • Uses Probability Mass Function (PMF)
  • \(P(X = x)\) makes sense

Examples: Dice rolls, number of emails, quiz scores

Continuous Random Variables

  • Uncountable values (infinite possibilities)
  • No gaps - any value in an interval
  • Uses Probability Density Function (PDF)
  • \(P(X = x) = 0\) for any specific value!

Examples: Height, weight, time, temperature

Why P(X = x) = 0 for Continuous Variables?

For continuous random variables, the probability of any exact value is zero!

Think about it: What’s the probability someone is exactly 5.7324681… feet tall?

Instead, we ask:

  • P(5.7 ≤ X ≤ 5.8)?

  • P(X ≤ 6.0)?

  • P(X > 5.5)?

Key insight: We calculate probabilities for intervals, not exact points.

Click to see why P(X = exact value) = 0

Probability Density Function (PDF)

🎯 Definition: The Probability Density Function (PDF) of a continuous random variable \(X\) is a function \(f(x)\) such that:

\[P(a \leq X \leq b) = \int_a^b f(x) \, dx\]

Properties of PDF:

  • \(f(x) \geq 0\) for all \(x\)
  • \(\int_{-\infty}^{\infty} f(x) \, dx = 1\)
  • \(f(x)\) is NOT a probability - it’s a density!

Key Insight:

The area under the PDF curve between \(a\) and \(b\) gives the probability that \(X\) falls in that interval.

Interactive PDF Demo: Understanding Density

Probability = Area under curve: Select an interval to see probability

Cumulative Distribution Function (CDF)

For continuous random variables, the CDF is:

\[F(x) = P(X \leq x) = \int_{-\infty}^x f(t) \, dt\]

Key relationship: \[f(x) = \frac{d}{dx}F(x)\]

The PDF is the derivative of the CDF!

Click anywhere to see F(x) = P(X ≤ x) for that point

Expected Value and Variance

Expected Value: \[E[X] = \mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx\]

Variance:

\[\text{Var}(X) = \sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) \, dx = E[X^2] - (E[X])^2\]

Where:

\[E[X^2] = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx\]

Important

Notice: Integrals replace sums when moving from discrete to continuous!

Common Continuous Distributions

Uniform Distribution

All values equally likely in an interval

Parameters: \(a\) (min), \(b\) (max)

PDF: \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\)

Mean: \(\frac{a+b}{2}\)

Variance: \(\frac{(b-a)^2}{12}\)

Use: Random numbers, waiting times

Normal Distribution

Bell-shaped, symmetric

Parameters: \(\mu\) (mean), \(\sigma^2\) (variance)

PDF: \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

Mean: \(\mu\)

Variance: \(\sigma^2\)

Use: Heights, test scores, errors

Exponential Distribution

Models waiting times

Parameters: \(\lambda\) (rate)

PDF: \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\)

Mean: \(\frac{1}{\lambda}\)

Variance: \(\frac{1}{\lambda^2}\)

Use: Time between events, lifetimes

The Normal Distribution

Why Normal is Special

  • Central Limit Theorem: Sample means approach normal
  • 68-95-99.7 Rule:
    • 68% within \(1 \sigma\) of \(\mu\)
    • 95% within \(2 \sigma\) of \(\mu\)
    • 99.7% within \(3 \sigma\) of \(\mu\)
  • Standard Normal: \(\mu = 0\) , \(\sigma = 1\)

Z-Score Transformation

\[Z = \frac{X - \mu}{\sigma}\]

Practice Problem 1: Uniform Distribution

A bus arrives uniformly between 10:00 AM and 10:20 AM. Let \(X\) = arrival time in minutes after 10:00 AM.

(a) What is the PDF of \(X\)?
(b) What’s the probability the bus arrives between 10:05 and 10:12?
(c) What’s the expected arrival time?

Solution. (a) \(X \sim \text{Uniform}(0, 20)\) \[f(x) = \frac{1}{20-0} = \frac{1}{20} \text{ for } 0 \leq x \leq 20\]

(b) \(P(5 \leq X \leq 12) = \int_5^{12} \frac{1}{20} dx = \frac{1}{20} \times (12-5) = \frac{7}{20} = 0.35\)

(c) \(E[X] = \frac{a+b}{2} = \frac{0+20}{2} = 10\) minutes after 10:00 AM

Practice Problem 2: Normal Distribution

Heights of adult women are normally distributed with μ = 64 inches and σ = 2.5 inches.

(a) What’s the probability a woman is taller than 67 inches?
(b) What height represents the 90th percentile?
(c) What’s the probability a woman is between 62 and 66 inches tall?

Solution. (a) \(P(X > 67) = P\left(Z > \frac{67-64}{2.5}\right) = P(Z > 1.2) = 1 - 0.8849 = 0.1151\)

(b) For 90th percentile: \(P(X \leq x) = 0.90\)
\(z_{0.90} = 1.28\), so \(x = 64 + 1.28(2.5) = 67.2\) inches

(c) \(P(62 \leq X \leq 66) = P\left(\frac{62-64}{2.5} \leq Z \leq \frac{66-64}{2.5}\right)\)
\(= P(-0.8 \leq Z \leq 0.8) = 0.7881 - 0.2119 = 0.5762\)

Practice Problem 3: Exponential Distribution

The time between customer arrivals at a store follows an exponential distribution with an average of 5 minutes between arrivals.

(a) What is the PDF?
(b) What’s the probability the next customer arrives within 3 minutes?
(c) What’s the probability no customer arrives in the next 10 minutes?

Solution. (a) Average = 5 minutes = \(\frac{1}{\lambda}\), so \(\lambda = 0.2\)
\[f(x) = 0.2e^{-0.2x} \text{ for } x \geq 0\]

(b) \(P(X \leq 3) = \int_0^3 0.2e^{-0.2x} dx = 1 - e^{-0.2 \times 3} = 1 - e^{-0.6} = 0.4512\)

(c) \(P(X > 10) = e^{-0.2 \times 10} = e^{-2} = 0.1353\)

Central Limit Theorem

Interactive CLT Demo: Sample Means Approach Normal

Population Distribution
Sample Means Distribution

CLT in Action: Run simulation to see the magic!

Transformations of Random Variables

Linear Transformations

If \(Y = aX + b\), then:

  • \(E[Y] = aE[X] + b\)
  • \(\text{Var}(Y) = a^2\text{Var}(X)\)
  • If \(X \sim N(\mu, \sigma^2)\), then \(Y \sim N(a\mu + b, a^2\sigma^2)\)

Standardization

\[Z = \frac{X - \mu}{\sigma} \sim N(0, 1)\]

Important: Normal distributions are closed under linear transformations!

Comparing Discrete and Continuous

Property Discrete Continuous
Probability Function PMF: \(P(X = x)\) PDF: \(f(x)\)
Exact Value Probability \(P(X = x) > 0\) possible \(P(X = x) = 0\) always
Interval Probability \(\sum P(X = x_i)\) \(\int_a^b f(x) dx\)
Expected Value \(\sum x \cdot P(X = x)\) \(\int x \cdot f(x) dx\)
Variance \(\sum (x-\mu)^2 P(X = x)\) \(\int (x-\mu)^2 f(x) dx\)
CDF \(\sum_{x_i \leq x} P(X = x_i)\) \(\int_{-\infty}^x f(t) dt\)

Properties of Continuous Distributions

Key Properties

  1. Memoryless Property (Exponential only):
    \(P(X > s+t | X > s) = P(X > t)\)

  2. Symmetry (Normal):
    \(P(X \leq \mu - a) = P(X \geq \mu + a)\)

  3. Scaling Invariance (Normal):
    Linear combinations of normals are normal

Useful Relationships

  • CDF to PDF: \(f(x) = F'(x)\)
  • PDF to CDF: \(F(x) = \int_{-\infty}^x f(t) dt\)
  • Complementary CDF: \(P(X > x) = 1 - F(x)\)

Remember: Area under PDF = 1, but PDF values can exceed 1!

Key Takeaways

Main Concepts

  • Continuous variables require PDFs, not PMFs
  • Probabilities are areas under curves, not function values
  • Integration replaces summation for continuous distributions
  • Normal distribution is central due to CLT

Distribution Selection

Choose distributions based on the data characteristics:

  • Uniform for equally likely intervals
  • Normal for symmetric, bell-shaped data
  • Exponential for waiting times/lifetimes
  • Use CLT when working with sample means

Key Principle

  • Central Limit Theorem makes normal distributions ubiquitous in statistics

Looking Ahead

Next Lecture: Statistical Inference

Topics we’ll cover:

  • Sampling distributions
  • Confidence intervals
  • Hypothesis testing
  • p-values and significance

Connection: Continuous distributions (especially normal) form the foundation for statistical inference

Questions?

Office Hours: 11AM on Thursday (link on Canvas)

Email: nmathlouthi@ucsb.edu

Next Class: Statistical Inference and Hypothesis Testing

Resources

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